Лекции по гидрогазодинамике
Подождите немного. Документ загружается.
cos (
π
2
− 2γ) = sin 2γ =
d~r
1
0
d~r
2
0
|d~r
1
0
||d~r
2
0
|
.
2γ A
sin 2γ ∼ 2γ; |d~r
1
0
| ∼ |d~r
1
|; |d~r
2
0
| ' |d~r
2
|.
2γ =
(d~r
1
+ (d~r
1
∇)~v) (d~r
2
+ (d~r
2
∇)~v)
|d~r
1
||d~r
2
|
.
~e
1
~e
2
d~r
1
= ~e
1
|d~r
1
|, d~r
2
= ~e
2
|d~r
2
|, d~r
1
d~r
2
= ~e
1
~e
2
= 0
2γ = ~e
1
(~e
2
∇)~v + ~e
2
(~e
1
∇)~v.
~e
1
~e
2
~
i,
~
j,
~
k ≡ ~e
1
, ~e
2
, ~e
3
2γ =
∂v
i
∂x
k
+
∂v
k
∂x
i
!
.
d~r
1
, d~r
2
, d~r
3
dV
dV = d~r
1
[d~r
2
× d~r
3
] .
~e
1
, ~e
2
, ~e
3
dV = ~e
1
[~e
2
× ~e
3
] |d~r
1
||d~r
2
||d~r
3
|.
0
d~r
1
0
, d~r
2
0
, d~r
3
0
dV dV
0
∂ δV
∂t
=
dV
0
− dV
dV
.
~
e
0
1
≈ ~e
1
,
~
e
0
2
≈ ~e
2
,
~
e
0
3
≈ ~e
3
.
d
~
r
0
k
= d~r
k
+ (d~r
k
∇)~v
∂ δV
∂t
=
(~e
1
∇)~v [~e
2
× ~e
3
] + (~e
2
∇)~v [~e
3
× ~e
1
] + (~e
3
∇)~v [~e
1
× ~e
2
]
~e
1
[~e
2
× ~e
3
]
.
~e
1
, ~e
2
, ~e
3
~e
1
, ~e
2
, ~e
3
≡
~
i,
~
j,
~
k,
∂ δV
∂t
=
∂v
x
∂x
+
∂v
y
∂y
+
∂v
z
∂z
= div ~v.
div ~v
1
V
dV
dt
= div ~v.
V = 1/ρ
dρ
dt
+ ρ div ~v = 0.
∂v
i
∂x
k
, (i, k = 1, 2, 3),
∂v
i
∂x
k
=
1
2
·
∂v
i
∂x
k
+
∂v
k
∂x
i
!
+
1
2
∂v
i
∂x
k
−
∂v
k
∂x
i
!
.
˙ε
˙ε =
∂v
x
∂x
1
2
∂v
x
∂y
+
∂v
y
∂x
!
1
2
∂v
x
∂z
+
∂v
z
∂x
!
1
2
∂v
y
∂x
+
∂v
x
∂y
!
∂v
y
∂y
1
2
∂v
y
∂z
+
∂v
z
∂y
!
1
2
∂v
z
∂x
+
∂v
x
∂z
!
1
2
∂v
z
∂y
+
∂v
y
∂z
!
∂v
z
∂z
.
1/2 rot(~v)
~ω
˙
Ω
˙
Ω =
0 ω
z
−ω
y
−ω
z
0 −ω
x
ω
y
ω
x
0
.
~v(~r + d~r, t) = ~v(~r, t) + [~ω × d~r ] +
˙
ε d~r.
2
1 = 1
2
.
1 = 1000 = 10
3
; 1 = 1000000 = 10
6
.
1 1
2
1 = 9, 8 ;
1 = 1
2
= 9.8 · 10
4
.
1 = 100000 = 10
5
.
. .
. ., . .
h = 1 = 10
−3
ρ ρ = 13600 /
3
1 . . .
p = ρ g h = 13600 · 9.8 · 10
−3
= 133 .
T = 4
C ρ = 1000 /
3
,
1 . . .
p = ρ g h = 1000 · 9.8 · 10
−3
= 9.8 .
=
2
. . . .
1 0.00001 1.02 · 10
−5
0.0075 1.02 · 10
−1
10
5
1 1.02 750 10.2
=
2
98100 0.981 1 735.5 10
. . 133.32 0.00133 0.00136 1 0.0136
. . 9810 0.0981 0.1 73.556 1
p > p
0
p = p = p − p
0
.
p = p
0
+ p .
p < p
0
p = p
0
− p.
p = p
0
− p .
∆p
∆V/V V
∆p = −E
∆V
V
,
−
∆p > 0 ∆V < 0
1
E
= β = −
1
V
∂V
∂p
=
1
ρ
∂ρ
∂p
.
β
β
T
= −
1
V
∂V
∂p
!
T
=
1
ρ
∂ρ
∂p
!
T
=
1
p
β
S
= −
1
V
∂V
∂p
!
S
=
1
ρ
∂ρ
∂p
!
S
=
1
k p
.
p V = p/ρ =
p V
k
= p/ρ
k
= k = C
p
/C
V
c
2
=
∂p
∂ρ
,
c
2
=
∂p
∂ρ
=
1
β ρ
=
E
ρ
.
c
T
=
v
u
u
t
p
ρ
=
√
R T ,
c
S
=
v
u
u
t
k p
ρ
=
√
k R T .
ρ
→ ∞ β = 0
1305
2060 1717
4464 20 1362
1275 50 1473
1275 10 1305
32373 1030
4
C
β
T
β
T
=
1
V
0
d V
d T
.
ρ =
ρ
0
1 + β
T
(T − T
0
)
,
ρ
0
T
0
p
ρ
= z R T,
p ρ R
T z
z
ρ
ρ
0
∆ρ = ρ −ρ
0
∆ρ
ρ
1.
ρ = ρ(p).
ρ = ρ
0
ρ ∼ ρ
0
+
∂ρ
∂p
!
ρ
0
,p
0
(p − p
0
).
∆p = (p − p
0
)
∆p
∆p ∼ ρ v
2
,